Properties of Family of 333,667 and Related Numbers/Squares

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Theorem

This page reports on certain properties, difficult to classify, of the number $333 \, 667$, and patterns arising.


The square of any number consisting of:

a string of $3$s

followed by:

a string of $6$s

followed by:

a single $7$

has its digits all in an increasing sequence:

\(\ds 333 \, 667^2\) \(=\) \(\ds 111 \, 333 \, 666 \, 889\)
\(\ds 33 \, 366 \, 667^2\) \(=\) \(\ds 1 \, 113 \, 334 \, 466 \, 688 \, 889\)


The same is true of numbers of the following form:

\(\ds 16 \, 667^2\) \(=\) \(\ds 277 \, 788 \, 889\)
\(\ds 333 \, 334^2\) \(=\) \(\ds 111 \, 111 \, 555 \, 556\)
\(\ds 333 \, 335^2\) \(=\) \(\ds 111 \, 112 \, 222 \, 225\)


Proof


This theorem requires a proof.
In particular: Straightforward but tedious exercise in induction and heavy algebra.
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Sources

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